Theorem riotaprop 6037

Description: Properties of a restricted definite description operator. Todo (df-riota 6011 update): can some uses of riota2f 6034 be shortened with this? (Contributed by NM, 23-Nov-2013.)

Hypotheses

Ref	Expression
riotaprop.0	|- F/ x ps
riotaprop.1	|- B = ( iota_ x e. A ph )
riotaprop.2	|- ( x = B -> ( ph <-> ps ) )

Assertion

Ref	Expression
riotaprop	|- ( E! x e. A ph -> ( B e. A /\ ps ) )

Distinct variable group:   x, A

Allowed substitution hints:   ph( x)   ps( x)   B( x)

Proof of Theorem riotaprop

Step	Hyp	Ref	Expression
1		riotaprop.1	. . 3 |- B = ( iota_ x e. A ph )
2		riotacl 6027	. . 3 |- ( E! x e. A ph -> ( iota_ x e. A ph ) e. A )
3	1, 2	eqeltrid 2321	. 2 |- ( E! x e. A ph -> B e. A )
4	1	eqcomi 2238	. . . 4 |- ( iota_ x e. A ph ) = B
5		nfriota1 6019	. . . . . 6 |- F/_ x ( iota_ x e. A ph )
6	1, 5	nfcxfr 2383	. . . . 5 |- F/_ x B
7		riotaprop.0	. . . . 5 |- F/ x ps
8		riotaprop.2	. . . . 5 |- ( x = B -> ( ph <-> ps ) )
9	6, 7, 8	riota2f 6034	. . . 4 |- ( ( B e. A /\ E! x e. A ph ) -> ( ps <-> ( iota_ x e. A ph ) = B ) )
10	4, 9	mpbiri 168	. . 3 |- ( ( B e. A /\ E! x e. A ph ) -> ps )
11	3, 10	mpancom 422	. 2 |- ( E! x e. A ph -> ps )
12	3, 11	jca 306	1 |- ( E! x e. A ph -> ( B e. A /\ ps ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1398   F/wnf 1509   e. wcel 2205   E!wreu 2524   iota_crio 6010

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2216

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-eu 2085  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-rex 2528  df-reu 2529  df-rab 2531  df-v 2817  df-sbc 3046  df-un 3218  df-in 3220  df-ss 3227  df-sn 3700  df-pr 3701  df-uni 3920  df-iota 5317  df-riota 6011

This theorem is referenced by:  lble  9238